Tverberg-type theory aims to establish sufficient conditions for a simplicial complex $Sigma$ such that every continuous map $fcolon Sigma to mathbb{R}^d$ maps $q$ points from pairwise disjoint faces to the same point in $mathbb{R}^d$. Such results are plentiful for $q$ a power of a prime. However, for $q$ with at least two distinct prime divisors, results that guarantee the existence of $q$-fold points of coincidence are non-existent---aside from immediate corollaries of the prime power case. Here we present a general method that yields such results beyond the case of prime powers. In particular, we prove previously conjectured upper bounds for the topological Tverberg problem for all $q$.
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